PRL 97, 092502 (2006)
PHYSICAL REVIEW LETTERS
week ending
1 SEPTEMBER 2006
Enhanced Effect of Temporal Variation of the Fine Structure Constant
and the Strong Interaction in 229 Th
V. V. Flambaum
School of Physics, The University of New South Wales, Sydney NSW 2052, Australia
(Received 24 April 2006; revised manuscript received 29 June 2006; published 31 August 2006)
The relative effects of the variation of the fine structure constant e2 =@c and the dimensionless
strong interaction parameter mq =QCD are enhanced by 5–6 orders of magnitude in a very narrow
ultraviolet transition between the ground and the first excited states in the 229 Th nucleus. It may be
possible to investigate this transition with laser spectroscopy. Such an experiment would have the potential
of improving the sensitivity to temporal variation of the fundamental constants by many orders of
magnitude.
DOI: 10.1103/PhysRevLett.97.092502
PACS numbers: 23.20.g, 06.20.Jr, 27.90.+b, 42.62.Fi
Unification theories applied to cosmology suggest a
possibility of variation of the fundamental constants in
the expanding Universe (see, e.g., the review in Ref. [1]).
There are hints of variation of and mq;e =QCD in quasar
absorption spectra, big bang nucleosynthesis, and Oklo
natural nuclear reactor data (see [2], and references
therein). Here QCD is the quantum chromodynamics
(QCD) scale, and mq and me are the quark and electron
masses, respectively. However, the majority of publications report only limits on possible variations (see, e.g.,
reviews in Refs. [1,3]). A very sensitive method to study
the variation in a laboratory consists of the comparison of
different optical and microwave atomic clocks (see recent
measurements in Refs. [4 –10]). An enhancement of the
relative effect of variation can be obtained in a transition
between almost degenerate levels in a Dy atom [11]. These
levels move in opposite directions if varies. An experiment is currently underway to place limits on variation
using this transition [12], but, unfortunately, one of the
levels has quite a large linewidth and this limits the accuracy. An enhancement of 1–3 orders exists in narrow
microwave molecular transitions [13]. Some atomic transitions with enhanced sensitivity are listed in Ref. [14].
A very narrow level 3:5 1 eV above the ground state
exists in the 229 Th nucleus [15] [in Ref. [16], the energy is
5:5 1 eV]. The position of this level was determined
from the energy differences of many high-energy transitions (between 25 and 320 KeV) to the ground and
excited states. The subtraction produces the large uncertainty in the position of the 3.5 eV excited state. The width
of this level is estimated to be about 104 Hz [17]. This
would explain why it is so hard to find the direct radiation
in this very weak transition. The direct measurements have
only given experimental limits on the width and energy of
this transition (see, e.g., [18]). A detailed discussion of the
measurements (including several unconfirmed claims of
the detection of the direct radiation) is presented in
Ref. [17]. However, the search for the direct radiation
continues [19]. The aim of the present work is to provide
a new strong motivation.
0031-9007=06=97(9)=092502(3)
The 229 Th transition is very narrow and can be investigated with laser spectroscopy. This makes 229 Th a possible reference for an optical clock of very high accuracy
and opens a new possibility for a laboratory search for the
variation of the fundamental constants [20]. Below, I will
show that there is an additional very important advantage.
The relative effects of variation of and mq =QCD are
enhanced by 5–6 orders of magnitude.
Note that there are other narrow low-energy levels in
nuclei, e.g., the 76 eV level in 235 U with the 26.6 min
lifetime (see, e.g., [20]). One may expect a similar enhancement there. Unfortunately, this level cannot be
reached with usual lasers. In principle, it may be investigated using a free-electron laser or synchrotron radiation.
However, the accuracy of the frequency measurements is
much lower in this case.
The ground state of the 229 Th nucleus is JP Nnz 5=2 633; i.e., the deformed oscillator quantum numbers
are N 6, nz 3, the projection of the valence neutron
orbital angular momentum on the nuclear symmetry axis
(internal z axis) is 3, the spin projection 1=2,
and the total angular momentum and the total angular
momentum projection are J 5=2. The
3.5 eV excited state is JP Nnz 3=2 631; i.e., it
has the same N 6 and nz 3. The values 1, 1=2, and J 3=2 are different. The energy of both
states may be described by an equation [21] E E0 C D2 ; i.e., the energy difference between the excited and ground state is ! Ee Eg 2C 8D. The
values of the constants C and D are presented, for example,
in Ref. [21]. Note that ! is 5 orders of magnitude smaller
than C and D. Therefore, for consistency of this simple
valence model, we must take 2C 8D. Based on the data
from [21], we will use the following numbers: 2C 8D 1:4 MeV. The relative variation of the transition frequency may be presented as
092502-1
2C 8D
D C
!
0:4 106
:
!
D
C
!
(1)
© 2006 The American Physical Society
PRL 97, 092502 (2006)
PHYSICAL REVIEW LETTERS
The large factor here appeared from the ratio 2C=! 8D=! 0:4 106 for ! 3:5 eV. The orbit-axis interaction constant D vanishes for zero deformation parameter 2 . Therefore, we should assume that
D const V0 2 , where V0 is the depth of the strong
potential. The nuclear deformation reduces the energy of
the Coulomb repulsion between the protons. Without this
repulsion, the deformation parameter 2 would probably
be zero [22]. Therefore, it is natural to assume that 2 const . Thus, we have D const V0 and
V0 D
:
V0
D
(2)
To estimate variation of V0 , we will use the Walecka model
[23], where the strong nuclear potential is produced by the
sigma and the omega meson exchanges
V
g2s erm g2v erm!
:
4 r
4 r
(3)
Using Eq. (3), we can find the depth of the potential well
[24,25]
V0 2
3
gs
g2v
:
4r30 m2 m2!
(4)
Here r0 1:2 fm is an internucleon distance. Note that the
nuclear potential in this model is a highly tuned small
difference of two large terms. Therefore, the contribution
of the variation of r0 is not as important as the contribution
of the meson mass variation, which is enhanced due to the
cancellation of two terms in V0 . The result is [24]
V0
m
m!
8:6
6:6
:
V0
m
m!
(5)
The final result will depend on the variation of the dimensionless parameter mq =QCD . During the following calculations, we shall assume that QCD does not vary and so we
shall speak about the variation of masses (this means that
we measure masses in units of QCD ). We shall restore the
explicit appearance of QCD in the final answers. The
dependence of the meson masses on the current light quark
mass mq mu md =2 has been calculated in Ref. [26]:
m =m 0:013mq =mq , m! =m! 0:034mq =mq .
This gives us
mq
V0
0:11
:
V0
mq
(6)
The relatively small contribution of the light quark mass is
explained by the fact that mq 5 MeV is very small. The
contribution of the strange quark mass ms 120 MeV
may be much larger. According to the calculation in
Ref. [24], m =m 0:54ms =ms , m! =m! week ending
1 SEPTEMBER 2006
0:15ms =ms , and V0 =V0 3:5ms =ms . By adding
all contributions, we obtain
mq
ms
D
0:11
3:5
:
m
ms
D
q
(7)
The reason for the enhancement here (5 times) is the
cancellation of the and ! meson contributions to V0 [see
Eq. (4)], which appears in the denominator of the relative
variation of D: D
D V0 =V0 . . . . However, the and
! mesons contribute with equal sign to the spin-orbit
interaction constant C [27]. Therefore, there is no ‘‘cancellation’’ enhancement here. However, there is another
efficient mechanism. The spin-orbit interaction is inversely
proportional to the nucleon mass MN squared, C / 1=MN2 ,
C
C 2MN =MN . The nucleon mass depends on the
quark masses: MN =MN Kq mq =mq Ks ms =ms ,
where Kq 0:045 and Ks 0:19 in Refs. [24,28], Kq 0:037 and Ks 0:011 in Ref. [29], and Kq 0:064 in
Ref. [26]. All three values of Kq are close to the average
value Kq 0:05. However, different methods of calculations give very different values of Ks . Fortunately, this is
not important, since the strange mass dependence of the meson is much stronger than that of the proton [due to the
SU(3) symmetry, contains a valence ss pair, another factor is strong repulsion of from the close K K , K 0 K 0 ,
states [24]]; also, there is the cancellation enhancement
of the contribution to D. As a result, we have
mq
m
D C 0:11 0:10
3:5 0:2 s :
C
mq
ms
D
(8)
The final estimate for the relative variation of the
transition frequency in Eq. (1) is
Xq
Xs 3:5 eV
!
5
;
10 4
10
!
Xq
Xs
!
229
Th
(9)
where Xq mq =QCD and Xs ms =QCD . When obtaining this estimate, we made several assumptions [the
Walecka model for the nuclear forces, linear dependence
of the deformation parameter on , and the SU(3) model
for the meson]. We plan to do a more sophisticated
numerical calculation which is not based on these assumptions. Unfortunately, it is quite complicated, and one
should not expect the result soon. However, it cannot
change the most important conclusion: There is a 5–6order enhancement in the relative variation of the transition
frequency.
To measure the variation of the fundamental constants,
one should do repeated measurements of the ratio of the
229
Th frequency to a frequency of any other narrow optical
or microwave transition (e.g., to the frequency standard,
the Cs microwave clock). The variation of the reference
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PHYSICAL REVIEW LETTERS
frequency can be neglected in comparison with the enhanced variation of the 229 Th frequency. Current atomic
clock limits on the variation of the fundamental constants
are approaching 1015 per year. Therefore, even without
any improvement of the measurement accuracy, one can
bring the sensitivity to the variation of the fundamental
constants to better than 1020 per year. Another advantage
is that the width of this nuclear transition is several orders
of magnitude smaller than a typical atomic clock width
(1–100 Hz). In principle, this may give another few
orders of magnitude improvement. We conclude that the
229 Th transition has an enormous potential for a laboratory
search for the variation of the fundamental constants.
This work is supported by the Australian Research
Council.
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